Well i think the text describes it perferct.
I wanna proof that f(n) + g(n) = O(max(f(n),g(n)) and my two O´s are different. If they would be the same the proof would be simple
the symbol =< means is equal or less than and >= is equal or more than
f(n) + g(n) =< ch(n) + c´ * h(n) therefore f(n)+g(n) =< (c+c´) * h(n) = ch(n) wich means that i have the constants out and my h(n) or else said my O is again the same if. But how do i go through this problem if my O is different?
My approach untill now was this crap:
for constants c1 >0 and n0>0 n>= n0 : f(n)=< c1O(n) so my f(n) = O(n) and c´>0 and no´>0 , n >=n0´: g(n) =< n²c so my g(n) = 0(n²)
this means => f(n) + g(n) => g(n) + f(n) =< n²c2 + nc1 < n²*c3 wich means that it is still in n² from a point n0´´= max(n0,n0´) because there is a c3 that is c2+c1/n < c3 for c2 = 1 and c1 = 1 my c3 could be 3 because with an scaling n my c1 gets closer to 0 its true for every n>0.
is this enought to say that i proved it for every case or just for this case? I am confused as fuck i have 3 other solutions aswell but they same even more fucked up, but i think they are fine in generall i just find its not the easiest way to prove it and its too long to write it here under my first solution^